Question 2 of 7: Cesium Chloride — Structure, Density, and X-ray Verification
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exam 04-BS-11, Properties of Materials — May 2017. 3 hours, closed-book examination (approved Casio or Sharp calculator only). Any five of the seven questions constitute a complete paper (only the first five in the answer book are marked); all questions are of equal value. All seven questions are solved below for completeness.
Reference texts: Callister & Rethwisch, Materials Science and Engineering: An Introduction, 9th ed. (dislocations and slip / Schmid’s law, mechanical behaviour, ionic crystal geometry and the radius-ratio rule, X-ray diffraction, diffusion and Fick’s laws, polymer molecular weight, ASTM grain size, strengthening and annealing, steel heat treatment and hardenability, casting defects, ceramics, glasses and composites).
Part (a) — coordination number, structure, lattice constant
Given. $r_{Cs^+}=167$ pm, $r_{Cl^-}=181$ pm; both ions monovalent (1:1 stoichiometry).
Find. Coordination number (CN), the Bravais/structure type, and the cubic lattice constant $a$.
Fig. Q2(a) — The CsCl structure: Cl⁻ ions on the eight cube corners (a simple-cubic lattice) with one Cs⁺ ion at the body centre. Cation and anion touch along the body diagonal, $a\sqrt3=2(r_++r_-)$.
Approach. Use the radius ratio to read the coordination number, identify the corresponding structure, then apply the touching condition for that geometry.
Radius ratio → coordination number. $$\frac{r_{Cs^+}}{r_{Cl^-}}=\frac{167}{181}=0.923.$$ Since $0.732\le0.923\le1.0$, the stable coordination is $\boxed{\text{CN}=8}$ (cubic, each ion surrounded by eight of the other).
Structure type. CN 8 with a 1:1 ratio is the CsCl structure: Cl⁻ at the eight corners forms a simple cubic lattice with Cs⁺ at the body centre. It is not BCC — the corner and centre sites hold different ions, so the lattice is primitive (simple) cubic with a two-ion basis.
Lattice constant. With CN 8 the cation and anion touch along the body diagonal, whose length $a\sqrt3$ equals $2(r_++r_-)$: $$a=\frac{2(r_{Cs^+}+r_{Cl^-})}{\sqrt3}=\frac{2(167+181)}{\sqrt3}=\boxed{402\ \text{pm}}=0.402\ \text{nm}.$$
Part (b) — density
Given. $a=402$ pm $=4.018\times10^{-8}$ cm; one Cs⁺ and one Cl⁻ per cell; $M_{Cs}=132.9$, $M_{Cl}=35.5$ g·mol$^{-1}$; $N_A=0.602\times10^{24}=6.02\times10^{23}$ mol$^{-1}$.
Formula mass per cell. One formula unit CsCl per cell: $M=132.9+35.5=168.4$ g·mol$^{-1}$.
Density. $$\rho=\frac{nM}{a^3N_A}=\frac{(1)(168.4)}{(4.018\times10^{-8})^3(6.02\times10^{23})}=\boxed{4.31\ \text{g}\cdot\text{cm}^{-3}}.$$ (The handbook value is $\approx3.99$ g·cm$^{-3}$; our result is a little high because the sum of ionic radii gives a slightly tighter cell than the measured one.)
Part (c) — X-ray verification
In a diffractometer, monochromatic X-rays of wavelength $\lambda$ strike the powder and the detector sweeps through $2\theta$, recording the angles at which constructive interference (Bragg peaks) occurs. Each peak obeys Bragg’s law $n\lambda=2d_{hkl}\sin\theta$, and for a cubic crystal $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$.
Fig. Q2(c)-i — Diffractometer geometry: the specimen is bathed in a collimated X-ray beam and the detector records intensity versus $2\theta$. Peaks appear where $n\lambda=2d\sin\theta$.
To verify part (a): measure the peak angles, convert each to a $d$-spacing via Bragg’s law, and index them (find integer $h,k,l$ with $a=d\sqrt{h^2+k^2+l^2}$ constant). A consistent $a\approx0.40$ nm confirms the lattice constant. The pattern of intensities confirms the CsCl (simple-cubic) structure: because the lattice is primitive, every reflection is allowed — there are no systematic absences. The structure factor $F=f_{Cs}+f_{Cl}\,(-1)^{h+k+l}$ makes reflections with $h+k+l$ even strong ($f_{Cs}+f_{Cl}$) and those with $h+k+l$ odd weak ($f_{Cs}-f_{Cl}$), but none vanish. This alternating strong/weak sequence — rather than the missing odd-sum lines of a true BCC metal — is the diffraction fingerprint of the CsCl structure.
Fig. Q2(c)-ii — Expected CsCl powder pattern: all reflections present, with $h+k+l$ even lines strong and $h+k+l$ odd lines weak (none extinguished).
